src/HOL/Cardinals/Wellorder_Embedding_Base.thy
author wenzelm
Wed, 10 Oct 2012 15:17:40 +0200
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permissions -rw-r--r--
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(*  Title:      HOL/Cardinals/Wellorder_Embedding_Base.thy
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    Author:     Andrei Popescu, TU Muenchen
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    Copyright   2012
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Well-order embeddings (base).
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*)
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header {* Well-Order Embeddings (Base) *}
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theory Wellorder_Embedding_Base
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imports "~~/src/HOL/Library/Zorn" Fun_More_Base Wellorder_Relation_Base
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begin
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text{* In this section, we introduce well-order {\em embeddings} and {\em isomorphisms} and
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prove their basic properties.  The notion of embedding is considered from the point
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of view of the theory of ordinals, and therefore requires the source to be injected
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as an {\em initial segment} (i.e., {\em order filter}) of the target.  A main result
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of this section is the existence of embeddings (in one direction or another) between
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any two well-orders, having as a consequence the fact that, given any two sets on
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any two types, one is smaller than (i.e., can be injected into) the other. *}
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subsection {* Auxiliaries *}
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lemma UNION_inj_on_ofilter:
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assumes WELL: "Well_order r" and
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        OF: "\<And> i. i \<in> I \<Longrightarrow> wo_rel.ofilter r (A i)" and
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       INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
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shows "inj_on f (\<Union> i \<in> I. A i)"
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proof-
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  have "wo_rel r" using WELL by (simp add: wo_rel_def)
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  hence "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i <= A j \<or> A j <= A i"
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  using wo_rel.ofilter_linord[of r] OF by blast
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  with WELL INJ show ?thesis
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  by (auto simp add: inj_on_UNION_chain)
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qed
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lemma under_underS_bij_betw:
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assumes WELL: "Well_order r" and WELL': "Well_order r'" and
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        IN: "a \<in> Field r" and IN': "f a \<in> Field r'" and
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        BIJ: "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
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shows "bij_betw f (rel.under r a) (rel.under r' (f a))"
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proof-
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  have "a \<notin> rel.underS r a \<and> f a \<notin> rel.underS r' (f a)"
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  unfolding rel.underS_def by auto
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  moreover
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  {have "Refl r \<and> Refl r'" using WELL WELL'
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   by (auto simp add: order_on_defs)
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   hence "rel.under r a = rel.underS r a \<union> {a} \<and>
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          rel.under r' (f a) = rel.underS r' (f a) \<union> {f a}"
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   using IN IN' by(auto simp add: rel.Refl_under_underS)
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  }
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  ultimately show ?thesis
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  using BIJ notIn_Un_bij_betw[of a "rel.underS r a" f "rel.underS r' (f a)"] by auto
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qed
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subsection {* (Well-order) embeddings, strict embeddings, isomorphisms and order-compatible
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functions  *}
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text{* Standardly, a function is an embedding of a well-order in another if it injectively and
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order-compatibly maps the former into an order filter of the latter.
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Here we opt for a more succinct definition (operator @{text "embed"}),
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asking that, for any element in the source, the function should be a bijection
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between the set of strict lower bounds of that element
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and the set of strict lower bounds of its image.  (Later we prove equivalence with
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the standard definition -- lemma @{text "embed_iff_compat_inj_on_ofilter"}.)
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A {\em strict embedding} (operator @{text "embedS"})  is a non-bijective embedding
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and an isomorphism (operator @{text "iso"}) is a bijective embedding.   *}
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definition embed :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
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where
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"embed r r' f \<equiv> \<forall>a \<in> Field r. bij_betw f (rel.under r a) (rel.under r' (f a))"
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lemmas embed_defs = embed_def embed_def[abs_def]
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text {* Strict embeddings: *}
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definition embedS :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
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where
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"embedS r r' f \<equiv> embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
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lemmas embedS_defs = embedS_def embedS_def[abs_def]
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definition iso :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
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where
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"iso r r' f \<equiv> embed r r' f \<and> bij_betw f (Field r) (Field r')"
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lemmas iso_defs = iso_def iso_def[abs_def]
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definition compat :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
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where
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"compat r r' f \<equiv> \<forall>a b. (a,b) \<in> r \<longrightarrow> (f a, f b) \<in> r'"
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lemma compat_wf:
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assumes CMP: "compat r r' f" and WF: "wf r'"
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shows "wf r"
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proof-
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  have "r \<le> inv_image r' f"
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  unfolding inv_image_def using CMP
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  by (auto simp add: compat_def)
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  with WF show ?thesis
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  using wf_inv_image[of r' f] wf_subset[of "inv_image r' f"] by auto
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qed
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lemma id_embed: "embed r r id"
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diff changeset
   120
by(auto simp add: id_def embed_def bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   121
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   122
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   123
lemma id_iso: "iso r r id"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   124
by(auto simp add: id_def embed_def iso_def bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   125
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   126
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   127
lemma embed_in_Field:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   128
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   129
        EMB: "embed r r' f" and IN: "a \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   130
shows "f a \<in> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   131
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   132
  have Well: "wo_rel r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   133
  using WELL by (auto simp add: wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   134
  hence 1: "Refl r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   135
  by (auto simp add: wo_rel.REFL)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   136
  hence "a \<in> rel.under r a" using IN rel.Refl_under_in by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   137
  hence "f a \<in> rel.under r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   138
  using EMB IN by (auto simp add: embed_def bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   139
  thus ?thesis unfolding Field_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   140
  by (auto simp: rel.under_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   141
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   142
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   143
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   144
lemma comp_embed:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   145
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   146
        EMB: "embed r r' f" and EMB': "embed r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   147
shows "embed r r'' (f' o f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   148
proof(unfold embed_def, auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   149
  fix a assume *: "a \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   150
  hence "bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   151
  using embed_def[of r] EMB by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   152
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   153
  {have "f a \<in> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   154
   using EMB WELL * by (auto simp add: embed_in_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   155
   hence "bij_betw f' (rel.under r' (f a)) (rel.under r'' (f' (f a)))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   156
   using embed_def[of r'] EMB' by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   157
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   158
  ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   159
  show "bij_betw (f' \<circ> f) (rel.under r a) (rel.under r'' (f'(f a)))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   160
  by(auto simp add: bij_betw_trans)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   161
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   162
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   163
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   164
lemma comp_iso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   165
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   166
        EMB: "iso r r' f" and EMB': "iso r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   167
shows "iso r r'' (f' o f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   168
using assms unfolding iso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   169
by (auto simp add: comp_embed bij_betw_trans)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   170
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   171
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   172
text{* That @{text "embedS"} is also preserved by function composition shall be proved only later.  *}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   173
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   174
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   175
lemma embed_Field:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   176
"\<lbrakk>Well_order r; embed r r' f\<rbrakk> \<Longrightarrow> f`(Field r) \<le> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   177
by (auto simp add: embed_in_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   178
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   179
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   180
lemma embed_preserves_ofilter:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   181
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   182
        EMB: "embed r r' f" and OF: "wo_rel.ofilter r A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   183
shows "wo_rel.ofilter r' (f`A)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   184
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   185
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   186
  from WELL have Well: "wo_rel r" unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   187
  from WELL' have Well': "wo_rel r'" unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   188
  from OF have 0: "A \<le> Field r" by(auto simp add: Well wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   189
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   190
  show ?thesis  using Well' WELL EMB 0 embed_Field[of r r' f]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   191
  proof(unfold wo_rel.ofilter_def, auto simp add: image_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   192
    fix a b'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   193
    assume *: "a \<in> A" and **: "b' \<in> rel.under r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   194
    hence "a \<in> Field r" using 0 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   195
    hence "bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   196
    using * EMB by (auto simp add: embed_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   197
    hence "f`(rel.under r a) = rel.under r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   198
    by (simp add: bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   199
    with ** image_def[of f "rel.under r a"] obtain b where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   200
    1: "b \<in> rel.under r a \<and> b' = f b" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   201
    hence "b \<in> A" using Well * OF
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   202
    by (auto simp add: wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   203
    with 1 show "\<exists>b \<in> A. b' = f b" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   204
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   205
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   206
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   207
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   208
lemma embed_Field_ofilter:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   209
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   210
        EMB: "embed r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   211
shows "wo_rel.ofilter r' (f`(Field r))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   212
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   213
  have "wo_rel.ofilter r (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   214
  using WELL by (auto simp add: wo_rel_def wo_rel.Field_ofilter)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   215
  with WELL WELL' EMB
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   216
  show ?thesis by (auto simp add: embed_preserves_ofilter)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   217
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   218
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   219
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   220
lemma embed_compat:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   221
assumes EMB: "embed r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   222
shows "compat r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   223
proof(unfold compat_def, clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   224
  fix a b
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   225
  assume *: "(a,b) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   226
  hence 1: "b \<in> Field r" using Field_def[of r] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   227
  have "a \<in> rel.under r b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   228
  using * rel.under_def[of r] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   229
  hence "f a \<in> rel.under r' (f b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   230
  using EMB embed_def[of r r' f]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   231
        bij_betw_def[of f "rel.under r b" "rel.under r' (f b)"]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   232
        image_def[of f "rel.under r b"] 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   233
  thus "(f a, f b) \<in> r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   234
  by (auto simp add: rel.under_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   235
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   236
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   237
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   238
lemma embed_inj_on:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   239
assumes WELL: "Well_order r" and EMB: "embed r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   240
shows "inj_on f (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   241
proof(unfold inj_on_def, clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   242
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   243
  from WELL have Well: "wo_rel r" unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   244
  with wo_rel.TOTAL[of r]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   245
  have Total: "Total r" by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   246
  from Well wo_rel.REFL[of r]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   247
  have Refl: "Refl r" by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   248
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   249
  fix a b
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   250
  assume *: "a \<in> Field r" and **: "b \<in> Field r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   251
         ***: "f a = f b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   252
  hence 1: "a \<in> Field r \<and> b \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   253
  unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   254
  {assume "(a,b) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   255
   hence "a \<in> rel.under r b \<and> b \<in> rel.under r b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   256
   using Refl by(auto simp add: rel.under_def refl_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   257
   hence "a = b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   258
   using EMB 1 ***
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   259
   by (auto simp add: embed_def bij_betw_def inj_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   260
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   261
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   262
  {assume "(b,a) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   263
   hence "a \<in> rel.under r a \<and> b \<in> rel.under r a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   264
   using Refl by(auto simp add: rel.under_def refl_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   265
   hence "a = b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   266
   using EMB 1 ***
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   267
   by (auto simp add: embed_def bij_betw_def inj_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   268
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   269
  ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   270
  show "a = b" using Total 1
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   271
  by (auto simp add: total_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   272
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   273
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   274
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   275
lemma embed_underS:
49753
a344f1a21211 eliminated spurious fact duplicates;
wenzelm
parents: 49310
diff changeset
   276
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
48975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   277
        EMB: "embed r r' f" and IN: "a \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   278
shows "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   279
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   280
  have "bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   281
  using assms by (auto simp add: embed_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   282
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   283
  {have "f a \<in> Field r'" using assms  embed_Field[of r r' f] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   284
   hence "rel.under r a = rel.underS r a \<union> {a} \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   285
          rel.under r' (f a) = rel.underS r' (f a) \<union> {f a}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   286
   using assms by (auto simp add: order_on_defs rel.Refl_under_underS)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   287
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   288
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   289
  {have "a \<notin> rel.underS r a \<and> f a \<notin> rel.underS r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   290
   unfolding rel.underS_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   291
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   292
  ultimately show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   293
  by (auto simp add: notIn_Un_bij_betw3)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   294
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   295
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   296
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   297
lemma embed_iff_compat_inj_on_ofilter:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   298
assumes WELL: "Well_order r" and WELL': "Well_order r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   299
shows "embed r r' f = (compat r r' f \<and> inj_on f (Field r) \<and> wo_rel.ofilter r' (f`(Field r)))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   300
using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   301
proof(auto simp add: embed_compat embed_inj_on embed_Field_ofilter,
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   302
      unfold embed_def, auto) (* get rid of one implication *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   303
  fix a
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   304
  assume *: "inj_on f (Field r)" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   305
         **: "compat r r' f" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   306
         ***: "wo_rel.ofilter r' (f`(Field r))" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   307
         ****: "a \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   308
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   309
  have Well: "wo_rel r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   310
  using WELL wo_rel_def[of r] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   311
  hence Refl: "Refl r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   312
  using wo_rel.REFL[of r] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   313
  have Total: "Total r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   314
  using Well wo_rel.TOTAL[of r] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   315
  have Well': "wo_rel r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   316
  using WELL' wo_rel_def[of r'] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   317
  hence Antisym': "antisym r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   318
  using wo_rel.ANTISYM[of r'] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   319
  have "(a,a) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   320
  using **** Well wo_rel.REFL[of r]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   321
        refl_on_def[of _ r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   322
  hence "(f a, f a) \<in> r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   323
  using ** by(auto simp add: compat_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   324
  hence 0: "f a \<in> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   325
  unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   326
  have "f a \<in> f`(Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   327
  using **** by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   328
  hence 2: "rel.under r' (f a) \<le> f`(Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   329
  using Well' *** wo_rel.ofilter_def[of r' "f`(Field r)"] by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   330
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   331
  show "bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   332
  proof(unfold bij_betw_def, auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   333
    show  "inj_on f (rel.under r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   334
    using *
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   335
    by (auto simp add: rel.under_Field subset_inj_on)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   336
  next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   337
    fix b assume "b \<in> rel.under r a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   338
    thus "f b \<in> rel.under r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   339
    unfolding rel.under_def using **
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   340
    by (auto simp add: compat_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   341
  next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   342
    fix b' assume *****: "b' \<in> rel.under r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   343
    hence "b' \<in> f`(Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   344
    using 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   345
    with Field_def[of r] obtain b where
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   346
    3: "b \<in> Field r" and 4: "b' = f b" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   347
    have "(b,a): r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   348
    proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   349
      {assume "(a,b) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   350
       with ** 4 have "(f a, b'): r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   351
       by (auto simp add: compat_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   352
       with ***** Antisym' have "f a = b'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   353
       by(auto simp add: rel.under_def antisym_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   354
       with 3 **** 4 * have "a = b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   355
       by(auto simp add: inj_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   356
      }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   357
      moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   358
      {assume "a = b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   359
       hence "(b,a) \<in> r" using Refl **** 3
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   360
       by (auto simp add: refl_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   361
      }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   362
      ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   363
      show ?thesis using Total **** 3 by (fastforce simp add: total_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   364
    qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   365
    with 4 show  "b' \<in> f`(rel.under r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   366
    unfolding rel.under_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   367
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   368
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   369
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   370
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   371
lemma inv_into_ofilter_embed:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   372
assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   373
        BIJ: "\<forall>b \<in> A. bij_betw f (rel.under r b) (rel.under r' (f b))" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   374
        IMAGE: "f ` A = Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   375
shows "embed r' r (inv_into A f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   376
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   377
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   378
  have Well: "wo_rel r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   379
  using WELL wo_rel_def[of r] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   380
  have Refl: "Refl r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   381
  using Well wo_rel.REFL[of r] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   382
  have Total: "Total r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   383
  using Well wo_rel.TOTAL[of r] by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   384
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   385
  have 1: "bij_betw f A (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   386
  proof(unfold bij_betw_def inj_on_def, auto simp add: IMAGE)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   387
    fix b1 b2
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   388
    assume *: "b1 \<in> A" and **: "b2 \<in> A" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   389
           ***: "f b1 = f b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   390
    have 11: "b1 \<in> Field r \<and> b2 \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   391
    using * ** Well OF by (auto simp add: wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   392
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   393
    {assume "(b1,b2) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   394
     hence "b1 \<in> rel.under r b2 \<and> b2 \<in> rel.under r b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   395
     unfolding rel.under_def using 11 Refl
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   396
     by (auto simp add: refl_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   397
     hence "b1 = b2" using BIJ * ** ***
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   398
     by (auto simp add: bij_betw_def inj_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   399
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   400
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   401
     {assume "(b2,b1) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   402
     hence "b1 \<in> rel.under r b1 \<and> b2 \<in> rel.under r b1"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   403
     unfolding rel.under_def using 11 Refl
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   404
     by (auto simp add: refl_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   405
     hence "b1 = b2" using BIJ * ** ***
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   406
     by (auto simp add: bij_betw_def inj_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   407
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   408
    ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   409
    show "b1 = b2"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   410
    using Total by (auto simp add: total_on_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   411
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   412
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   413
  let ?f' = "(inv_into A f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   414
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   415
  have 2: "\<forall>b \<in> A. bij_betw ?f' (rel.under r' (f b)) (rel.under r b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   416
  proof(clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   417
    fix b assume *: "b \<in> A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   418
    hence "rel.under r b \<le> A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   419
    using Well OF by(auto simp add: wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   420
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   421
    have "f ` (rel.under r b) = rel.under r' (f b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   422
    using * BIJ by (auto simp add: bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   423
    ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   424
    show "bij_betw ?f' (rel.under r' (f b)) (rel.under r b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   425
    using 1 by (auto simp add: bij_betw_inv_into_subset)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   426
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   427
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   428
  have 3: "\<forall>b' \<in> Field r'. bij_betw ?f' (rel.under r' b') (rel.under r (?f' b'))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   429
  proof(clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   430
    fix b' assume *: "b' \<in> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   431
    have "b' = f (?f' b')" using * 1
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   432
    by (auto simp add: bij_betw_inv_into_right)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   433
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   434
    {obtain b where 31: "b \<in> A" and "f b = b'" using IMAGE * by force
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   435
     hence "?f' b' = b" using 1 by (auto simp add: bij_betw_inv_into_left)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   436
     with 31 have "?f' b' \<in> A" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   437
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   438
    ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   439
    show  "bij_betw ?f' (rel.under r' b') (rel.under r (?f' b'))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   440
    using 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   441
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   442
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   443
  thus ?thesis unfolding embed_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   444
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   445
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   446
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   447
lemma inv_into_underS_embed:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   448
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   449
        BIJ: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   450
        IN: "a \<in> Field r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   451
        IMAGE: "f ` (rel.underS r a) = Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   452
shows "embed r' r (inv_into (rel.underS r a) f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   453
using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   454
by(auto simp add: wo_rel_def wo_rel.underS_ofilter inv_into_ofilter_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   455
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   456
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   457
lemma inv_into_Field_embed:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   458
assumes WELL: "Well_order r" and EMB: "embed r r' f" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   459
        IMAGE: "Field r' \<le> f ` (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   460
shows "embed r' r (inv_into (Field r) f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   461
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   462
  have "(\<forall>b \<in> Field r. bij_betw f (rel.under r b) (rel.under r' (f b)))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   463
  using EMB by (auto simp add: embed_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   464
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   465
  have "f ` (Field r) \<le> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   466
  using EMB WELL by (auto simp add: embed_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   467
  ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   468
  show ?thesis using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   469
  by(auto simp add: wo_rel_def wo_rel.Field_ofilter inv_into_ofilter_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   470
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   471
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   472
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   473
lemma inv_into_Field_embed_bij_betw:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   474
assumes WELL: "Well_order r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   475
        EMB: "embed r r' f" and BIJ: "bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   476
shows "embed r' r (inv_into (Field r) f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   477
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   478
  have "Field r' \<le> f ` (Field r)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   479
  using BIJ by (auto simp add: bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   480
  thus ?thesis using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   481
  by(auto simp add: inv_into_Field_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   482
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   483
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   484
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   485
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   486
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   487
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   488
subsection {* Given any two well-orders, one can be embedded in the other *}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   489
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   490
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   491
text{* Here is an overview of the proof of of this fact, stated in theorem
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   492
@{text "wellorders_totally_ordered"}:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   493
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   494
   Fix the well-orders @{text "r::'a rel"} and @{text "r'::'a' rel"}.
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   495
   Attempt to define an embedding @{text "f::'a \<Rightarrow> 'a'"} from @{text "r"} to @{text "r'"} in the
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   496
   natural way by well-order recursion ("hoping" that @{text "Field r"} turns out to be smaller
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   497
   than @{text "Field r'"}), but also record, at the recursive step, in a function
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   498
   @{text "g::'a \<Rightarrow> bool"}, the extra information of whether @{text "Field r'"}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   499
   gets exhausted or not.
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   500
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   501
   If @{text "Field r'"} does not get exhausted, then @{text "Field r"} is indeed smaller
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   502
   and @{text "f"} is the desired embedding from @{text "r"} to @{text "r'"}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   503
   (lemma @{text "wellorders_totally_ordered_aux"}).
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   504
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   505
   Otherwise, it means that @{text "Field r'"} is the smaller one, and the inverse of
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   506
   (the "good" segment of) @{text "f"} is the desired embedding from @{text "r'"} to @{text "r"}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   507
   (lemma @{text "wellorders_totally_ordered_aux2"}).
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   508
*}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   509
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   510
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   511
lemma wellorders_totally_ordered_aux:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   512
fixes r ::"'a rel"  and r'::"'a' rel" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   513
      f :: "'a \<Rightarrow> 'a'" and a::'a
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   514
assumes WELL: "Well_order r" and WELL': "Well_order r'" and IN: "a \<in> Field r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   515
        IH: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   516
        NOT: "f ` (rel.underS r a) \<noteq> Field r'" and SUC: "f a = wo_rel.suc r' (f`(rel.underS r a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   517
shows "bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   518
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   519
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   520
  have Well: "wo_rel r" using WELL unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   521
  hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   522
  have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   523
  have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   524
  have OF: "wo_rel.ofilter r (rel.underS r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   525
  by (auto simp add: Well wo_rel.underS_ofilter)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   526
  hence UN: "rel.underS r a = (\<Union>  b \<in> rel.underS r a. rel.under r b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   527
  using Well wo_rel.ofilter_under_UNION[of r "rel.underS r a"] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   528
  (* Gather facts about elements of rel.underS r a *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   529
  {fix b assume *: "b \<in> rel.underS r a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   530
   hence t0: "(b,a) \<in> r \<and> b \<noteq> a" unfolding rel.underS_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   531
   have t1: "b \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   532
   using * rel.underS_Field[of r a] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   533
   have t2: "f`(rel.under r b) = rel.under r' (f b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   534
   using IH * by (auto simp add: bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   535
   hence t3: "wo_rel.ofilter r' (f`(rel.under r b))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   536
   using Well' by (auto simp add: wo_rel.under_ofilter)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   537
   have "f`(rel.under r b) \<le> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   538
   using t2 by (auto simp add: rel.under_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   539
   moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   540
   have "b \<in> rel.under r b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   541
   using t1 by(auto simp add: Refl rel.Refl_under_in)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   542
   ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   543
   have t4:  "f b \<in> Field r'" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   544
   have "f`(rel.under r b) = rel.under r' (f b) \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   545
         wo_rel.ofilter r' (f`(rel.under r b)) \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   546
         f b \<in> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   547
   using t2 t3 t4 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   548
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   549
  hence bFact:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   550
  "\<forall>b \<in> rel.underS r a. f`(rel.under r b) = rel.under r' (f b) \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   551
                       wo_rel.ofilter r' (f`(rel.under r b)) \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   552
                       f b \<in> Field r'" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   553
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   554
  have subField: "f`(rel.underS r a) \<le> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   555
  using bFact by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   556
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   557
  have OF': "wo_rel.ofilter r' (f`(rel.underS r a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   558
  proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   559
    have "f`(rel.underS r a) = f`(\<Union>  b \<in> rel.underS r a. rel.under r b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   560
    using UN by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   561
    also have "\<dots> = (\<Union>  b \<in> rel.underS r a. f`(rel.under r b))" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   562
    also have "\<dots> = (\<Union>  b \<in> rel.underS r a. (rel.under r' (f b)))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   563
    using bFact by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   564
    finally
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   565
    have "f`(rel.underS r a) = (\<Union>  b \<in> rel.underS r a. (rel.under r' (f b)))" .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   566
    thus ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   567
    using Well' bFact
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   568
          wo_rel.ofilter_UNION[of r' "rel.underS r a" "\<lambda> b. rel.under r' (f b)"] by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   569
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   570
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   571
  have "f`(rel.underS r a) \<union> rel.AboveS r' (f`(rel.underS r a)) = Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   572
  using Well' OF' by (auto simp add: wo_rel.ofilter_AboveS_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   573
  hence NE: "rel.AboveS r' (f`(rel.underS r a)) \<noteq> {}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   574
  using subField NOT by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   575
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   576
  have INCL1: "f`(rel.underS r a) \<le> rel.underS r' (f a) "
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   577
  proof(auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   578
    fix b assume *: "b \<in> rel.underS r a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   579
    have "f b \<noteq> f a \<and> (f b, f a) \<in> r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   580
    using subField Well' SUC NE *
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   581
          wo_rel.suc_greater[of r' "f`(rel.underS r a)" "f b"] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   582
    thus "f b \<in> rel.underS r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   583
    unfolding rel.underS_def by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   584
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   585
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   586
  have INCL2: "rel.underS r' (f a) \<le> f`(rel.underS r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   587
  proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   588
    fix b' assume "b' \<in> rel.underS r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   589
    hence "b' \<noteq> f a \<and> (b', f a) \<in> r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   590
    unfolding rel.underS_def by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   591
    thus "b' \<in> f`(rel.underS r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   592
    using Well' SUC NE OF'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   593
          wo_rel.suc_ofilter_in[of r' "f ` rel.underS r a" b'] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   594
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   595
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   596
  have INJ: "inj_on f (rel.underS r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   597
  proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   598
    have "\<forall>b \<in> rel.underS r a. inj_on f (rel.under r b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   599
    using IH by (auto simp add: bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   600
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   601
    have "\<forall>b. wo_rel.ofilter r (rel.under r b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   602
    using Well by (auto simp add: wo_rel.under_ofilter)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   603
    ultimately show  ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   604
    using WELL bFact UN
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   605
          UNION_inj_on_ofilter[of r "rel.underS r a" "\<lambda>b. rel.under r b" f]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   606
    by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   607
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   608
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   609
  have BIJ: "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   610
  unfolding bij_betw_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   611
  using INJ INCL1 INCL2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   612
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   613
  have "f a \<in> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   614
  using Well' subField NE SUC
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   615
  by (auto simp add: wo_rel.suc_inField)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   616
  thus ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   617
  using WELL WELL' IN BIJ under_underS_bij_betw[of r r' a f] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   618
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   619
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   620
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   621
lemma wellorders_totally_ordered_aux2:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   622
fixes r ::"'a rel"  and r'::"'a' rel" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   623
      f :: "'a \<Rightarrow> 'a'" and g :: "'a \<Rightarrow> bool"  and a::'a
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   624
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   625
MAIN1:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   626
  "\<And> a. (False \<notin> g`(rel.underS r a) \<and> f`(rel.underS r a) \<noteq> Field r'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   627
          \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   628
         \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   629
         (\<not>(False \<notin> (g`(rel.underS r a)) \<and> f`(rel.underS r a) \<noteq> Field r')
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   630
          \<longrightarrow> g a = False)" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   631
MAIN2: "\<And> a. a \<in> Field r \<and> False \<notin> g`(rel.under r a) \<longrightarrow>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   632
              bij_betw f (rel.under r a) (rel.under r' (f a))" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   633
Case: "a \<in> Field r \<and> False \<in> g`(rel.under r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   634
shows "\<exists>f'. embed r' r f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   635
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   636
  have Well: "wo_rel r" using WELL unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   637
  hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   638
  have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   639
  have Antisym: "antisym r" using Well wo_rel.ANTISYM[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   640
  have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   641
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   642
  have 0: "rel.under r a = rel.underS r a \<union> {a}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   643
  using Refl Case by(auto simp add: rel.Refl_under_underS)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   644
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   645
  have 1: "g a = False"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   646
  proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   647
    {assume "g a \<noteq> False"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   648
     with 0 Case have "False \<in> g`(rel.underS r a)" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   649
     with MAIN1 have "g a = False" by blast}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   650
    thus ?thesis by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   651
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   652
  let ?A = "{a \<in> Field r. g a = False}"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   653
  let ?a = "(wo_rel.minim r ?A)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   654
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   655
  have 2: "?A \<noteq> {} \<and> ?A \<le> Field r" using Case 1 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   656
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   657
  have 3: "False \<notin> g`(rel.underS r ?a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   658
  proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   659
    assume "False \<in> g`(rel.underS r ?a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   660
    then obtain b where "b \<in> rel.underS r ?a" and 31: "g b = False" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   661
    hence 32: "(b,?a) \<in> r \<and> b \<noteq> ?a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   662
    by (auto simp add: rel.underS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   663
    hence "b \<in> Field r" unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   664
    with 31 have "b \<in> ?A" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   665
    hence "(?a,b) \<in> r" using wo_rel.minim_least 2 Well by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   666
    (* again: why worked without type annotations? *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   667
    with 32 Antisym show False
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   668
    by (auto simp add: antisym_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   669
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   670
  have temp: "?a \<in> ?A"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   671
  using Well 2 wo_rel.minim_in[of r ?A] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   672
  hence 4: "?a \<in> Field r" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   673
  (*   *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   674
  have 5: "g ?a = False" using temp by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   675
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   676
  have 6: "f`(rel.underS r ?a) = Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   677
  using MAIN1[of ?a] 3 5 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   678
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   679
  have 7: "\<forall>b \<in> rel.underS r ?a. bij_betw f (rel.under r b) (rel.under r' (f b))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   680
  proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   681
    fix b assume as: "b \<in> rel.underS r ?a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   682
    moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   683
    have "wo_rel.ofilter r (rel.underS r ?a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   684
    using Well by (auto simp add: wo_rel.underS_ofilter)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   685
    ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   686
    have "False \<notin> g`(rel.under r b)" using 3 Well by (auto simp add: wo_rel.ofilter_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   687
    moreover have "b \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   688
    unfolding Field_def using as by (auto simp add: rel.underS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   689
    ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   690
    show "bij_betw f (rel.under r b) (rel.under r' (f b))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   691
    using MAIN2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   692
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   693
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   694
  have "embed r' r (inv_into (rel.underS r ?a) f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   695
  using WELL WELL' 7 4 6 inv_into_underS_embed[of r ?a f r'] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   696
  thus ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   697
  unfolding embed_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   698
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   699
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   700
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   701
theorem wellorders_totally_ordered:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   702
fixes r ::"'a rel"  and r'::"'a' rel"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   703
assumes WELL: "Well_order r" and WELL': "Well_order r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   704
shows "(\<exists>f. embed r r' f) \<or> (\<exists>f'. embed r' r f')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   705
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   706
  (* Preliminary facts *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   707
  have Well: "wo_rel r" using WELL unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   708
  hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   709
  have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   710
  have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   711
  (* Main proof *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   712
  obtain H where H_def: "H =
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   713
  (\<lambda>h a. if False \<notin> (snd o h)`(rel.underS r a) \<and> (fst o h)`(rel.underS r a) \<noteq> Field r'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   714
                then (wo_rel.suc r' ((fst o h)`(rel.underS r a)), True)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   715
                else (undefined, False))" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   716
  have Adm: "wo_rel.adm_wo r H"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   717
  using Well
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   718
  proof(unfold wo_rel.adm_wo_def, clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   719
    fix h1::"'a \<Rightarrow> 'a' * bool" and h2::"'a \<Rightarrow> 'a' * bool" and x
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   720
    assume "\<forall>y\<in>rel.underS r x. h1 y = h2 y"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   721
    hence "\<forall>y\<in>rel.underS r x. (fst o h1) y = (fst o h2) y \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   722
                          (snd o h1) y = (snd o h2) y" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   723
    hence "(fst o h1)`(rel.underS r x) = (fst o h2)`(rel.underS r x) \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   724
           (snd o h1)`(rel.underS r x) = (snd o h2)`(rel.underS r x)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   725
    by (auto simp add: image_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   726
    thus "H h1 x = H h2 x" by (simp add: H_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   727
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   728
  (* More constant definitions:  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   729
  obtain h::"'a \<Rightarrow> 'a' * bool" and f::"'a \<Rightarrow> 'a'" and g::"'a \<Rightarrow> bool"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   730
  where h_def: "h = wo_rel.worec r H" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   731
        f_def: "f = fst o h" and g_def: "g = snd o h" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   732
  obtain test where test_def:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   733
  "test = (\<lambda> a. False \<notin> (g`(rel.underS r a)) \<and> f`(rel.underS r a) \<noteq> Field r')" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   734
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   735
  have *: "\<And> a. h a  = H h a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   736
  using Adm Well wo_rel.worec_fixpoint[of r H] by (simp add: h_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   737
  have Main1:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   738
  "\<And> a. (test a \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True) \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   739
         (\<not>(test a) \<longrightarrow> g a = False)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   740
  proof-  (* How can I prove this withou fixing a? *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   741
    fix a show "(test a \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True) \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   742
                (\<not>(test a) \<longrightarrow> g a = False)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   743
    using *[of a] test_def f_def g_def H_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   744
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   745
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   746
  let ?phi = "\<lambda> a. a \<in> Field r \<and> False \<notin> g`(rel.under r a) \<longrightarrow>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   747
                   bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   748
  have Main2: "\<And> a. ?phi a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   749
  proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   750
    fix a show "?phi a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   751
    proof(rule wo_rel.well_order_induct[of r ?phi],
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   752
          simp only: Well, clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   753
      fix a
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   754
      assume IH: "\<forall>b. b \<noteq> a \<and> (b,a) \<in> r \<longrightarrow> ?phi b" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   755
             *: "a \<in> Field r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   756
             **: "False \<notin> g`(rel.under r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   757
      have 1: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   758
      proof(clarify)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   759
        fix b assume ***: "b \<in> rel.underS r a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   760
        hence 0: "(b,a) \<in> r \<and> b \<noteq> a" unfolding rel.underS_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   761
        moreover have "b \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   762
        using *** rel.underS_Field[of r a] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   763
        moreover have "False \<notin> g`(rel.under r b)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   764
        using 0 ** Trans rel.under_incr[of r b a] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   765
        ultimately show "bij_betw f (rel.under r b) (rel.under r' (f b))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   766
        using IH by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   767
      qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   768
      (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   769
      have 21: "False \<notin> g`(rel.underS r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   770
      using ** rel.underS_subset_under[of r a] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   771
      have 22: "g`(rel.under r a) \<le> {True}" using ** by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   772
      moreover have 23: "a \<in> rel.under r a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   773
      using Refl * by (auto simp add: rel.Refl_under_in)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   774
      ultimately have 24: "g a = True" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   775
      have 2: "f`(rel.underS r a) \<noteq> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   776
      proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   777
        assume "f`(rel.underS r a) = Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   778
        hence "g a = False" using Main1 test_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   779
        with 24 show False using ** by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   780
      qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   781
      (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   782
      have 3: "f a = wo_rel.suc r' (f`(rel.underS r a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   783
      using 21 2 Main1 test_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   784
      (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   785
      show "bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   786
      using WELL  WELL' 1 2 3 *
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   787
            wellorders_totally_ordered_aux[of r r' a f] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   788
    qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   789
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   790
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   791
  let ?chi = "(\<lambda> a. a \<in> Field r \<and> False \<in> g`(rel.under r a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   792
  show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   793
  proof(cases "\<exists>a. ?chi a")
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   794
    assume "\<not> (\<exists>a. ?chi a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   795
    hence "\<forall>a \<in> Field r.  bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   796
    using Main2 by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   797
    thus ?thesis unfolding embed_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   798
  next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   799
    assume "\<exists>a. ?chi a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   800
    then obtain a where "?chi a" by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   801
    hence "\<exists>f'. embed r' r f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   802
    using wellorders_totally_ordered_aux2[of r r' g f a]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   803
          WELL WELL' Main1 Main2 test_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   804
    thus ?thesis by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   805
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   806
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   807
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   808
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   809
subsection {* Uniqueness of embeddings  *}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   810
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   811
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   812
text{* Here we show a fact complementary to the one from the previous subsection -- namely,
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   813
that between any two well-orders there is {\em at most} one embedding, and is the one
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   814
definable by the expected well-order recursive equation.  As a consequence, any two
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   815
embeddings of opposite directions are mutually inverse. *}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   816
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   817
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   818
lemma embed_determined:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   819
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   820
        EMB: "embed r r' f" and IN: "a \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   821
shows "f a = wo_rel.suc r' (f`(rel.underS r a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   822
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   823
  have "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   824
  using assms by (auto simp add: embed_underS)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   825
  hence "f`(rel.underS r a) = rel.underS r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   826
  by (auto simp add: bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   827
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   828
  {have "f a \<in> Field r'" using IN
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   829
   using EMB WELL embed_Field[of r r' f] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   830
   hence "f a = wo_rel.suc r' (rel.underS r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   831
   using WELL' by (auto simp add: wo_rel_def wo_rel.suc_underS)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   832
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   833
  ultimately show ?thesis by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   834
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   835
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   836
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   837
lemma embed_unique:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   838
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   839
        EMBf: "embed r r' f" and EMBg: "embed r r' g"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   840
shows "a \<in> Field r \<longrightarrow> f a = g a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   841
proof(rule wo_rel.well_order_induct[of r], auto simp add: WELL wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   842
  fix a
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   843
  assume IH: "\<forall>b. b \<noteq> a \<and> (b,a): r \<longrightarrow> b \<in> Field r \<longrightarrow> f b = g b" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   844
         *: "a \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   845
  hence "\<forall>b \<in> rel.underS r a. f b = g b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   846
  unfolding rel.underS_def by (auto simp add: Field_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   847
  hence "f`(rel.underS r a) = g`(rel.underS r a)" by force
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   848
  thus "f a = g a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   849
  using assms * embed_determined[of r r' f a] embed_determined[of r r' g a] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   850
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   851
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   852
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   853
lemma embed_bothWays_inverse:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   854
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   855
        EMB: "embed r r' f" and EMB': "embed r' r f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   856
shows "(\<forall>a \<in> Field r. f'(f a) = a) \<and> (\<forall>a' \<in> Field r'. f(f' a') = a')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   857
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   858
  have "embed r r (f' o f)" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   859
  by(auto simp add: comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   860
  moreover have "embed r r id" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   861
  by (auto simp add: id_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   862
  ultimately have "\<forall>a \<in> Field r. f'(f a) = a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   863
  using assms embed_unique[of r r "f' o f" id] id_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   864
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   865
  {have "embed r' r' (f o f')" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   866
   by(auto simp add: comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   867
   moreover have "embed r' r' id" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   868
   by (auto simp add: id_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   869
   ultimately have "\<forall>a' \<in> Field r'. f(f' a') = a'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   870
   using assms embed_unique[of r' r' "f o f'" id] id_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   871
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   872
  ultimately show ?thesis by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   873
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   874
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   875
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   876
lemma embed_bothWays_bij_betw:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   877
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   878
        EMB: "embed r r' f" and EMB': "embed r' r g"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   879
shows "bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   880
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   881
  let ?A = "Field r"  let ?A' = "Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   882
  have "embed r r (g o f) \<and> embed r' r' (f o g)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   883
  using assms by (auto simp add: comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   884
  hence 1: "(\<forall>a \<in> ?A. g(f a) = a) \<and> (\<forall>a' \<in> ?A'. f(g a') = a')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   885
  using WELL id_embed[of r] embed_unique[of r r "g o f" id]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   886
        WELL' id_embed[of r'] embed_unique[of r' r' "f o g" id]
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   887
        id_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   888
  have 2: "(\<forall>a \<in> ?A. f a \<in> ?A') \<and> (\<forall>a' \<in> ?A'. g a' \<in> ?A)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   889
  using assms embed_Field[of r r' f] embed_Field[of r' r g] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   890
  (*  *)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   891
  show ?thesis
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   892
  proof(unfold bij_betw_def inj_on_def, auto simp add: 2)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   893
    fix a b assume *: "a \<in> ?A" "b \<in> ?A" and **: "f a = f b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   894
    have "a = g(f a) \<and> b = g(f b)" using * 1 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   895
    with ** show "a = b" by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   896
  next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   897
    fix a' assume *: "a' \<in> ?A'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   898
    hence "g a' \<in> ?A \<and> f(g a') = a'" using 1 2 by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   899
    thus "a' \<in> f ` ?A" by force
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   900
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   901
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   902
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   903
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   904
lemma embed_bothWays_iso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   905
assumes WELL: "Well_order r"  and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   906
        EMB: "embed r r' f" and EMB': "embed r' r g"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   907
shows "iso r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   908
unfolding iso_def using assms by (auto simp add: embed_bothWays_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   909
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   910
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   911
subsection {* More properties of embeddings, strict embeddings and isomorphisms  *}
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   912
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   913
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   914
lemma embed_bothWays_Field_bij_betw:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   915
assumes WELL: "Well_order r" and WELL': "Well_order r'" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   916
        EMB: "embed r r' f" and EMB': "embed r' r f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   917
shows "bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   918
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   919
  have "(\<forall>a \<in> Field r. f'(f a) = a) \<and> (\<forall>a' \<in> Field r'. f(f' a') = a')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   920
  using assms by (auto simp add: embed_bothWays_inverse)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   921
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   922
  have "f`(Field r) \<le> Field r' \<and> f' ` (Field r') \<le> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   923
  using assms by (auto simp add: embed_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   924
  ultimately
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   925
  show ?thesis using bij_betw_byWitness[of "Field r" f' f "Field r'"] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   926
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   927
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   928
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   929
lemma embedS_comp_embed:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   930
assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   931
        and  EMB: "embedS r r' f" and EMB': "embed r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   932
shows "embedS r r'' (f' o f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   933
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   934
  let ?g = "(f' o f)"  let ?h = "inv_into (Field r) ?g"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   935
  have 1: "embed r r' f \<and> \<not> (bij_betw f (Field r) (Field r'))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   936
  using EMB by (auto simp add: embedS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   937
  hence 2: "embed r r'' ?g"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   938
  using WELL EMB' comp_embed[of r r' f r'' f'] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   939
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   940
  {assume "bij_betw ?g (Field r) (Field r'')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   941
   hence "embed r'' r ?h" using 2 WELL
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   942
   by (auto simp add: inv_into_Field_embed_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   943
   hence "embed r' r (?h o f')" using WELL' EMB'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   944
   by (auto simp add: comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   945
   hence "bij_betw f (Field r) (Field r')" using WELL WELL' 1
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   946
   by (auto simp add: embed_bothWays_Field_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   947
   with 1 have False by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   948
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   949
  ultimately show ?thesis unfolding embedS_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   950
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   951
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   952
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   953
lemma embed_comp_embedS:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   954
assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   955
        and  EMB: "embed r r' f" and EMB': "embedS r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   956
shows "embedS r r'' (f' o f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   957
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   958
  let ?g = "(f' o f)"  let ?h = "inv_into (Field r) ?g"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   959
  have 1: "embed r' r'' f' \<and> \<not> (bij_betw f' (Field r') (Field r''))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   960
  using EMB' by (auto simp add: embedS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   961
  hence 2: "embed r r'' ?g"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   962
  using WELL EMB comp_embed[of r r' f r'' f'] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   963
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   964
  {assume "bij_betw ?g (Field r) (Field r'')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   965
   hence "embed r'' r ?h" using 2 WELL
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   966
   by (auto simp add: inv_into_Field_embed_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   967
   hence "embed r'' r' (f o ?h)" using WELL'' EMB
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   968
   by (auto simp add: comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   969
   hence "bij_betw f' (Field r') (Field r'')" using WELL' WELL'' 1
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   970
   by (auto simp add: embed_bothWays_Field_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   971
   with 1 have False by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   972
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   973
  ultimately show ?thesis unfolding embedS_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   974
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   975
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   976
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   977
lemma embed_comp_iso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   978
assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   979
        and  EMB: "embed r r' f" and EMB': "iso r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   980
shows "embed r r'' (f' o f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   981
using assms unfolding iso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   982
by (auto simp add: comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   983
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   984
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   985
lemma iso_comp_embed:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   986
assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   987
        and  EMB: "iso r r' f" and EMB': "embed r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   988
shows "embed r r'' (f' o f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   989
using assms unfolding iso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   990
by (auto simp add: comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   991
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   992
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   993
lemma embedS_comp_iso:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   994
assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   995
        and  EMB: "embedS r r' f" and EMB': "iso r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   996
shows "embedS r r'' (f' o f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   997
using assms unfolding iso_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   998
by (auto simp add: embedS_comp_embed)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
   999
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1000
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1001
lemma iso_comp_embedS:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1002
assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1003
        and  EMB: "iso r r' f" and EMB': "embedS r' r'' f'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1004
shows "embedS r r'' (f' o f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1005
using assms unfolding iso_def  using embed_comp_embedS
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1006
by (auto simp add: embed_comp_embedS)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1007
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1008
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1009
lemma embedS_Field:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1010
assumes WELL: "Well_order r" and EMB: "embedS r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1011
shows "f ` (Field r) < Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1012
proof-
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1013
  have "f`(Field r) \<le> Field r'" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1014
  by (auto simp add: embed_Field embedS_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1015
  moreover
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1016
  {have "inj_on f (Field r)" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1017
   by (auto simp add: embedS_def embed_inj_on)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1018
   hence "f`(Field r) \<noteq> Field r'" using EMB
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1019
   by (auto simp add: embedS_def bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1020
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1021
  ultimately show ?thesis by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1022
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1023
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1024
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1025
lemma embedS_iff:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1026
assumes WELL: "Well_order r" and ISO: "embed r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1027
shows "embedS r r' f = (f ` (Field r) < Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1028
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1029
  assume "embedS r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1030
  thus "f ` Field r \<subset> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1031
  using WELL by (auto simp add: embedS_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1032
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1033
  assume "f ` Field r \<subset> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1034
  hence "\<not> bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1035
  unfolding bij_betw_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1036
  thus "embedS r r' f" unfolding embedS_def
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1037
  using ISO by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1038
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1039
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1040
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1041
lemma iso_Field:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1042
"iso r r' f \<Longrightarrow> f ` (Field r) = Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1043
using assms by (auto simp add: iso_def bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1044
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1045
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1046
lemma iso_iff:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1047
assumes "Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1048
shows "iso r r' f = (embed r r' f \<and> f ` (Field r) = Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1049
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1050
  assume "iso r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1051
  thus "embed r r' f \<and> f ` (Field r) = Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1052
  by (auto simp add: iso_Field iso_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1053
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1054
  assume *: "embed r r' f \<and> f ` Field r = Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1055
  hence "inj_on f (Field r)" using assms by (auto simp add: embed_inj_on)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1056
  with * have "bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1057
  unfolding bij_betw_def by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1058
  with * show "iso r r' f" unfolding iso_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1059
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1060
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1061
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1062
lemma iso_iff2:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1063
assumes "Well_order r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1064
shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and>
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1065
                     (\<forall>a \<in> Field r. \<forall>b \<in> Field r.
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1066
                         (((a,b) \<in> r) = ((f a, f b) \<in> r'))))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1067
using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1068
proof(auto simp add: iso_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1069
  fix a b
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1070
  assume "embed r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1071
  hence "compat r r' f" using embed_compat[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1072
  moreover assume "(a,b) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1073
  ultimately show "(f a, f b) \<in> r'" using compat_def[of r] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1074
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1075
  let ?f' = "inv_into (Field r) f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1076
  assume "embed r r' f" and 1: "bij_betw f (Field r) (Field r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1077
  hence "embed r' r ?f'" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1078
  by (auto simp add: inv_into_Field_embed_bij_betw)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1079
  hence 2: "compat r' r ?f'" using embed_compat[of r'] by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1080
  fix a b assume *: "a \<in> Field r" "b \<in> Field r" and **: "(f a,f b) \<in> r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1081
  hence "?f'(f a) = a \<and> ?f'(f b) = b" using 1
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1082
  by (auto simp add: bij_betw_inv_into_left)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1083
  thus "(a,b) \<in> r" using ** 2 compat_def[of r' r ?f'] by fastforce
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1084
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1085
  assume *: "bij_betw f (Field r) (Field r')" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1086
         **: "\<forall>a\<in>Field r. \<forall>b\<in>Field r. ((a, b) \<in> r) = ((f a, f b) \<in> r')"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1087
  have 1: "\<And> a. rel.under r a \<le> Field r \<and> rel.under r' (f a) \<le> Field r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1088
  by (auto simp add: rel.under_Field)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1089
  have 2: "inj_on f (Field r)" using * by (auto simp add: bij_betw_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1090
  {fix a assume ***: "a \<in> Field r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1091
   have "bij_betw f (rel.under r a) (rel.under r' (f a))"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1092
   proof(unfold bij_betw_def, auto)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1093
     show "inj_on f (rel.under r a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1094
     using 1 2 by (auto simp add: subset_inj_on)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1095
   next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1096
     fix b assume "b \<in> rel.under r a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1097
     hence "a \<in> Field r \<and> b \<in> Field r \<and> (b,a) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1098
     unfolding rel.under_def by (auto simp add: Field_def Range_def Domain_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1099
     with 1 ** show "f b \<in> rel.under r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1100
     unfolding rel.under_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1101
   next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1102
     fix b' assume "b' \<in> rel.under r' (f a)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1103
     hence 3: "(b',f a) \<in> r'" unfolding rel.under_def by simp
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1104
     hence "b' \<in> Field r'" unfolding Field_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1105
     with * obtain b where "b \<in> Field r \<and> f b = b'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1106
     unfolding bij_betw_def by force
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1107
     with 3 ** ***
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1108
     show "b' \<in> f ` (rel.under r a)" unfolding rel.under_def by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1109
   qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1110
  }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1111
  thus "embed r r' f" unfolding embed_def using * by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1112
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1113
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1114
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1115
lemma iso_iff3:
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1116
assumes WELL: "Well_order r" and WELL': "Well_order r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1117
shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and> compat r r' f)"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1118
proof
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1119
  assume "iso r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1120
  thus "bij_betw f (Field r) (Field r') \<and> compat r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1121
  unfolding compat_def using WELL by (auto simp add: iso_iff2 Field_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1122
next
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1123
  have Well: "wo_rel r \<and> wo_rel r'" using WELL WELL'
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1124
  by (auto simp add: wo_rel_def)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1125
  assume *: "bij_betw f (Field r) (Field r') \<and> compat r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1126
  thus "iso r r' f"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1127
  unfolding "compat_def" using assms
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1128
  proof(auto simp add: iso_iff2)
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1129
    fix a b assume **: "a \<in> Field r" "b \<in> Field r" and
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1130
                  ***: "(f a, f b) \<in> r'"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1131
    {assume "(b,a) \<in> r \<or> b = a"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1132
     hence "(b,a): r"using Well ** wo_rel.REFL[of r] refl_on_def[of _ r] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1133
     hence "(f b, f a) \<in> r'" using * unfolding compat_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1134
     hence "f a = f b"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1135
     using Well *** wo_rel.ANTISYM[of r'] antisym_def[of r'] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1136
     hence "a = b" using * ** unfolding bij_betw_def inj_on_def by auto
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1137
     hence "(a,b) \<in> r" using Well ** wo_rel.REFL[of r] refl_on_def[of _ r] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1138
    }
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1139
    thus "(a,b) \<in> r"
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1140
    using Well ** wo_rel.TOTAL[of r] total_on_def[of _ r] by blast
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1141
  qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1142
qed
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1143
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1144
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1145
7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff changeset
  1146
end